A proof of the Grünbaum conjecture

Abstract

Let V be an n-dimensional real Banach space and let λ(V) denote its absolute projection constant. For any N ∈ N with N ≥ n define λ N n = sup{λ(V) : dim(V) = n, V ⊂ l (N) ∞ }, λn = sup{λ(V) : dim(V) = n}. A well-known Grünbaum conjecture [Trans. Amer. Math. Soc. 95 (1960)] says that λ2 = 4/3. König and Tomczak-Jaegermann [J. Funct. Anal. 119 (1994)] made an attempt to prove this conjecture. Unfortunately, their Proposition 3.1, used in the proof, is incorrect. In this paper a complete proof of the Grünbaum conjecture is presented.

Absolute Lipschitz extendability and linear projection constants

“…As pointed out in the introduction, ae(n) ≤ λ n−1 . Thus, since λ 2 = 4 3 , as established by Chalmers and Lewicki in [11], it follows that ae(3) ≤ 4 3 . Now, Lemma 4.1 tells us that for a three-point metric space X with d(x, x ) = 1 for all distinct points x, x ∈ X, one has ae(X) = 2 − 2 3 = 4 3 .…”

“…Due to an important result of Kadets and Snobar (see [23]), one has λ n ≤ √ n for all n ≥ 1. The maximal projection constant λ n is difficult to compute, the only known values are λ 1 = 1 and λ 2 = 4 3 , the former due to the Hahn-Banach theorem and the latter due to Chalmers and Lewicki [11]. In [26], König proved that λ n ≥ √ n − 1 for a subsequence of integers n. Hence, by taking into account Lee and Naor's upper bound of ae(n), we find that (1.9) is not sharp for n ≥ 1 large enough.…”

On the Unique Minimality of the Fourier Projection inLpSpaces

“…Then a 4 = 0. Now using the above equality, we obtain that a 0 = −a 4 Now assume that a 4 = 0. Then a 3 = 0.…”

Uniqueness of minimal projections in smooth expanded matrix spaces

“…In the theory of minimal projection three main problems are considered: existence and uniqueness of minimal projections [15][16][17][19][20][21][22][23][24][25][26][27][28][29] , finding estimates of the constant λ(T ; S) [2][3][4][5][7][8][9][10][11][12][13] and finding concrete formulas for minimal projections [6,9,18,24]. As one can see this theory is widely studied by many authors also recently [1,11,12,14,18,23].…”